#### Vol. 14, No. 3, 2021

 Recent Issues
 The Journal About the Journal Editorial Board Editors’ Interests Subscriptions Submission Guidelines Submission Form Policies for Authors Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Restriction of toral eigenfunctions to totally geodesic submanifolds

### Xiaoqi Huang and Cheng Zhang

Vol. 14 (2021), No. 3, 861–880
##### Abstract

We estimate the ${L}^{2}$ norm of the restriction to a totally geodesic submanifold of the eigenfunctions of the Laplace–Beltrami operator on the standard flat torus ${\mathbb{𝕋}}^{d}$, $d\ge 2$. We reduce getting correct bounds to counting lattice points in the intersection of some $\nu$-transverse bands on the sphere. Moreover, we prove the correct bounds for rational totally geodesic submanifolds of arbitrary codimension. In particular, we verify the conjecture of Bourgain–Rudnick on ${L}^{2}$-restriction estimates for rational hyperplanes. On ${\mathbb{𝕋}}^{2}$, we prove the uniform ${L}^{2}$ restriction bounds for closed geodesics. On ${\mathbb{𝕋}}^{3}$, we obtain explicit ${L}^{2}$ restriction estimates for the totally geodesic submanifolds, which improve the corresponding results by Burq–Gérard–Tzvetkov, Hu, and Chen–Sogge.

##### Keywords
eigenfunction estimates, restriction, geodesic
##### Mathematical Subject Classification 2010
Primary: 11P21, 35P20, 58J50